Theorem bd0 16969

Description: A formula equivalent to a bounded one is bounded. See also bd0r 16970. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bd0.min	⊢ BOUNDED 𝜑
bd0.maj	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
bd0	⊢ BOUNDED 𝜓

Proof of Theorem bd0

Step	Hyp	Ref	Expression
1		bd0.min	. 2 ⊢ BOUNDED 𝜑
2		bd0.maj	. . 3 ⊢ (𝜑 ↔ 𝜓)
3	2	ax-bd0 16958	. 2 ⊢ (BOUNDED 𝜑 → BOUNDED 𝜓)
4	1, 3	ax-mp 5	1 ⊢ BOUNDED 𝜓

This proof depends on syntax axioms:   ↔ wb 105  BOUNDED wbd 16957

This proof depends on axioms:  ax-mp 5  ax-bd0 16958

This theorem is used by:  bd0r  16970  bdth  16976  bdnth  16979  bdnthALT  16980  bdph  16995  bdsbc  17003  bdsnss  17018  bdcint  17022  bdeqsuc  17026  bdcriota  17028  bj-axun2  17060